The broad, long-term objective and specific aims is to establish the utility of using unrelated matched and partially mismatched cord blood cells as a source of hematopoietic stem cells for allogeneic grafting in adult patients with advanced malignant diseases. Cord blood has been used successfully to reconstitute pediatric patients following myeloablative preparatory regimens, but the utility of these cells in the adult populations remains untested. If the success encountered in the pediatric population could be expanded to adults, then the current difficulties with finding acceptable histocompatible donors could be overcome in the majority of patients. This research project will: 1.) Determine the percent and timing of engraftment in adult patients, 2.) observe for the incidence and outcome of acute and chronic graft-versus-host disease, 3.) analyze the immunological reconstitution of these cord blood cells in patients with and without complications following this procedure. The preliminary results suggests that engraftment can occur in adult patients treated for malignant diseases and that while the numbers of patients treated are small, the overall incidence of graft- versus-host disease (GVHD) is low given the degree of incompatibility at the major histocompatibility complex (MHC). At the end of the three years, the results will provide an indication of the percentage of time where engraftment does or does not occur and how it is related to other clinical factors such as the degree of mismatching at the MHC. Moreover, the overall incidence of GVHD will also be known. More importantly, the manner in which the immune system recovers following the establishment of hematopoiesis from this novel source will be delineated and correlation with clinical outcome will also be performed. It is expected that upon completion of these studies, a prospective randomized study will be conducted comparing the source of unrelated donor hematopoietic cells.